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Hilbert cube
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==The Hilbert cube as a metric space== It is sometimes convenient to think of the Hilbert cube as a [[metric space]], indeed as a specific subset of a separable [[Hilbert space]] (that is, a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not to think of it as a product of copies of <math>[0, 1],</math> but instead as <math display=block>[0, 1] \times [0, 1/2] \times [0, 1/3] \times \cdots;</math> as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an [[infinite sequence]] <math display=block>\left(x_n\right)_{n \in \N}</math> that satisfies <math display=block>0 \leq x_n \leq 1/n.</math> Any such sequence belongs to the Hilbert space [[Lp space#The p-norm in countably infinite dimensions|<math>\ell_2,</math>]] so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the [[product topology]] in the above definition.
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